The distinction between parametric and non parametric is as follows: Does the number of parameters in the model grows as the number of examples in the dataset grows?
Parametric Example: Linear/Ridge regression, fixed number of parameters regardless of the number of examples.
Non-Parametric Example: Tree-based models, the parameters are the indexes of split variables, split values, and leaves values. As the number of examples grows so does the tree height and thus the number of parameters.
ANN case: as the number of parameter grows so does the number of neurons/layers and therefore considered as non-parametric.